Comparative analysis of powerlaw type fin problem using wavelet collocation and Galerkin methods

20141117 https://doi.org/10.14419/ijamr.v3i4.3137 
Linear, nonlinear, differential equation, fins, wavelet Galerkin, collocation method. 
In this paper, Wavelet Collocation Method and Wavelet Galerkin Method have been used to evaluate the temperature distribution of a straight rectangular fin. The linear problem has been solved by Wavelet Galerkin Method while nonlinear problem by Wavelet Collocation Method. It has been observed that accuracy increases as the number of basis function increases. The result thus obtained is compared with other available results obtained by using approximate analytic methods such as Adom ian Decomposition Method, Differential Transformation Method as well as exact solution. It has been observed that the result obtained by present method is exactly same as that obtained by exact method. The method provides a unique solution forÂ n = 3/2, N= Â± 0.9 and n = 3, N = Â±0.4. The justification of unique solution gets confirmed from Figs. 7 and 8. The present method provides single solution for all existing values. The convergence analysis of the proposed method along side numerical procedure for this boundary value problem is given to test wider applicability and accuracy of the method.

References
 I.N. Dul'kin, G.I. Garas'ko, "Analysis of the 1D heat conduction problem for a single fin with temperature dependent heat transfer coefficient: Part I Extended inverse and direct solutions", International Journal of Heat and Mass Transfer, 51, (2008), pp.33093324.
 I.N. Dul'kin, G.I. Garas'ko, "Analytical solutions of the 1D heat", Int. J. of Heat and Mass Transfer, Vol.45(2), (2002), pp.18951903.
 A. Moradi, "Analytical solution for fin with temperature dependent heat transfer coefficient", Int. J. of Eng. and App. Science, Vol.3, issue 2, (2011), pp.112.
 MinHsing Chang, "A decomposition solution for fins with temperature dependent surface heat flux", Int. J. of Heat and Mass Transfer, 48 (2005), pp.18191824.
 M.S.H. Chowdhury, I.Hashim and O. Abdulaziz, "Comparison of homotopy analysis method and homotopyperturbation method for purely nonlinear fintype problems", Communications in Nonlinear Science and Numerical Simulation, 14, (2009), pp.371378.
 Rafael Cortell, "A numerical analysis to the nonlinear fin problem", J. of Zhejiang University Science, A 9 (5), (2008), pp.648653.
 S. Abbasbandy and E. Shivanian, "Exact analytical solution of a nonlinear equation arising in heat transfer", Physics Letters A, 374 (2010), pp.567574.
 Chang M.H., "A decomposition solution for fins with temperature dependent surface heat flux",Int. J. Heat Mass Transfer", 48 (2005), pp.18191824.
 K. N. Rai and K.D. Rai, "A Numerical Technique for the solution of Transport Problem", Proceeding, Fourth National Conference on Thermal Systems Dept. of Mech. Eng. ITBHU, (2003), pp.268277.
 F. Khani, M. Ahmadzadeh Raji, H. Hamedi Nejad, "Analytical solution and efficiency of the nonlinear fin problem with temperaturedependent thermal conductivity and heat transfer coefficient", Commun Nolinear Sci Numer Simulat, 14 (2009), pp.33273338.
 Sin Kim and ChengHung Huang, "A series solution of the nonlinear fin problem with temperature dependent thermal conductivity and heat transfer coefficient", J. Phys. D: Appl. Phys., 40, (2007), pp.29792987.
 Ebrahim Momoniat, "A comparison of two formulations of the fin efficiency for straight fins", Acta Mech. Sin., 28 (2), (2012), pp.444449.
 Ching  Huang Chiu and Cha'o Kuang Chen, "A decomposition method for solving the convective longitudinal fins with variable thermal conductivity", Int. J. of Heat and Mass Transfer, 45 (2002), pp.20672075.
 A Rajabi, "Homotopy perturbation method for fin efficiency of convective straight fins with temperaturedependent thermal conductivity", Physics Letters A, 364, (2007), pp.3337.
 Cihat Arslanturk, "A decomposition method for fin efficiency of convective straight fins with temperaturedependent thermal conductivity", Int. Commun. in Heat and Mass Transfer, 32(2005), pp.831841.
 G. Domairry and M. Fazeli, "Homotopy analysis method to determine the fin efficiency of convective straight fins with temperaturedependent thermal conductivity", Commun. in nonlinear Sci. and Nonlinear Simulation,14 (2009), pp.489499.
 D.D. Ganji, M.J. Hosseini and J. Shayegh, Some nonlinear heat transfer equations solved by three approximate methods, Int. Commun. in Heat and Mass Transfer, 34 (2007), pp.10031016.
 M. Razzaghi and S. Yousefi, "The Legendre wavelets operational matrix of integration", Int.J. of Systems Science, Vol.32, No.4, (2001), pp.495502.
 F. Mohammadi, M.M. Hosseini and syed Tauseef MohyudDin, "Legendre Wavelet Galerkin method for solving ordinary differential equations with nonanalytic solution", Int. J. of Sys. Sci.}, Vol. 42, No. 4, (2011), pp.579585.
 M.W. Frazier, "An Introduction to Wavelets Through Linear Algebra", SpringerVerlag New York Berlin Heidelberg, SPIN 10557627, (1999), pp.470480.

How to Cite
Singh, S., Upadhyay, S., & Rai, K. N. (2014). Comparative analysis of powerlaw type fin problem using wavelet collocation and Galerkin methods. International Journal of Applied Mathematical Research, 3(4), 534546. https://doi.org/10.14419/ijamr.v3i4.3137